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21
A Linear Logical Framework
, 1996
"... We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF c ..."
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Cited by 217 (44 self)
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We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of MiniML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cutelimination. 1 Introduction A logical framework is a formal system desig...
Higherorder logic programming
 HANDBOOK OF LOGIC IN AI AND LOGIC PROGRAMMING, VOLUME 5: LOGIC PROGRAMMING. OXFORD (1998
"... ..."
Higherorder Unification via Explicit Substitutions (Extended Abstract)
 Proceedings of LICS'95
, 1995
"... Higherorder unification is equational unification for βηconversion. But it is not firstorder equational unification, as substitution has to avoid capture. In this paper higherorder unification is reduced to firstorder equational unification in a suitable theory: the λσcal ..."
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Cited by 103 (13 self)
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Higherorder unification is equational unification for βηconversion. But it is not firstorder equational unification, as substitution has to avoid capture. In this paper higherorder unification is reduced to firstorder equational unification in a suitable theory: the λσcalculus of explicit substitutions.
Unification: A multidisciplinary survey
 ACM Computing Surveys
, 1989
"... The unification problem and several variants are presented. Various algorithms and data structures are discussed. Research on unification arising in several areas of computer science is surveyed, these areas include theorem proving, logic programming, and natural language processing. Sections of the ..."
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Cited by 103 (0 self)
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The unification problem and several variants are presented. Various algorithms and data structures are discussed. Research on unification arising in several areas of computer science is surveyed, these areas include theorem proving, logic programming, and natural language processing. Sections of the paper include examples that highlight particular uses
Equations and rewrite rules: a survey
 In Formal Language Theory: Perspectives and Open Problems
, 1980
"... bY ..."
HigherOrder Horn Clauses
 JOURNAL OF THE ACM
, 1990
"... A generalization of Horn clauses to a higherorder logic is described and examined as a basis for logic programming. In qualitative terms, these higherorder Horn clauses are obtained from the firstorder ones by replacing firstorder terms with simply typed #terms and by permitting quantification ..."
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Cited by 62 (19 self)
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A generalization of Horn clauses to a higherorder logic is described and examined as a basis for logic programming. In qualitative terms, these higherorder Horn clauses are obtained from the firstorder ones by replacing firstorder terms with simply typed #terms and by permitting quantification over all occurrences of function symbols and some occurrences of predicate symbols. Several prooftheoretic results concerning these extended clauses are presented. One result shows that although the substitutions for predicate variables can be quite complex in general, the substitutions necessary in the context of higherorder Horn clauses are tightly constrained. This observation is used to show that these higherorder formulas can specify computations in a fashion similar to firstorder Horn clauses. A complete theorem proving procedure is also described for the extension. This procedure is obtained by interweaving higherorder unification with backchaining and goal reductions, and constitutes a higherorder generalization of SLDresolution. These results have a practical realization in the higherorder logic programming language called λProlog.
Inducing Probabilistic CCG Grammars from Logical Form with HigherOrder Unification
"... This paper addresses the problem of learning to map sentences to logical form, given training data consisting of natural language sentences paired with logical representations of their meaning. Previous approaches have been designed for particular natural languages or specific meaning representation ..."
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Cited by 48 (9 self)
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This paper addresses the problem of learning to map sentences to logical form, given training data consisting of natural language sentences paired with logical representations of their meaning. Previous approaches have been designed for particular natural languages or specific meaning representations; here we present a more general method. The approach induces a probabilistic CCG grammar that represents the meaning of individual words and defines how these meanings can be combined to analyze complete sentences. We use higherorder unification to define a hypothesis space containing all grammars consistent with the training data, and develop an online learning algorithm that efficiently searches this space while simultaneously estimating the parameters of a loglinear parsing model. Experiments demonstrate high accuracy on benchmark data sets in four languages with two different meaning representations. 1
Extensions and Applications of Higherorder Unification
, 1990
"... ... unification problems. Then, in this framework, we develop a new unification algorithm for acalculus with dependent function (II) types. This algorithm is especially useful as it provides for mechanization in the very expressive Logical Framework (LF). The development (objectlanguages). The ric ..."
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Cited by 25 (1 self)
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... unification problems. Then, in this framework, we develop a new unification algorithm for acalculus with dependent function (II) types. This algorithm is especially useful as it provides for mechanization in the very expressive Logical Framework (LF). The development (objectlanguages). The rich structure of a typedcalculus,asopposedtotraditional,rst generalideaistouseacalculusasametalanguageforrepresentingvariousotherlanguages thelattercase,thealgorithmisincomplete,thoughstillquiteusefulinpractice. Thelastpartofthedissertationprovidesexamplesoftheusefulnessofthealgorithms.The algorithmrstfordependentproduct()types,andsecondforimplicitpolymorphism.In involvessignicantcomplicationsnotarisingHuet'scorrespondingalgorithmforthesimply orderabstractsyntaxtrees,allowsustoexpressrules,e.g.,programtransformationand typedcalculus,primarilybecauseitmustdealwithilltypedterms.Wethenextendthis Wecanthenuseunicationinthemetalanguagetomechanizeapplicationoftheserules.
Structuring and Automating Hardware Proofs in a HigherOrder TheoremProving Environment
 Formal Methods in System Design
, 1993
"... . In this article we present a structured approach to formal hardware verification by modelling circuits at the registertransfer level using a restricted form of higherorder logic. This restricted form of higherorder logic is sufficient for obtaining succinct descriptions of hierarchically design ..."
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Cited by 20 (7 self)
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. In this article we present a structured approach to formal hardware verification by modelling circuits at the registertransfer level using a restricted form of higherorder logic. This restricted form of higherorder logic is sufficient for obtaining succinct descriptions of hierarchically designed registertransfer circuits. By exploiting the structure of the underlying hardware proofs and limiting the form of descriptions used, we have attained nearly complete automation in proving the equivalences of the specifications and implementations. A hardwarespecific tool called MEPHISTO converts the original goal into a set of simpler subgoals, which are then automatically solved by a generalpurpose, firstorder prover called FAUST. Furthermore, the complete verification framework is being integrated within a commercial VLSI CAD framework. Keywords: hardware verification, higherorder logic 1 Introduction The past decade has witnessed the spiralling of interest within the academic com...
On Unification Problems in Restricted SecondOrder Languages
 In Annual Conf. of the European Ass. of Computer Science Logic (CSL98
, 1998
"... We review known results and improve known boundaries between the decidable and the undecidable cases of secondorder unification with various restrictions on secondorder variables. As a key tool we prove an undecidability result that provides a partial solution to an open problem about simultaneous ..."
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Cited by 6 (3 self)
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We review known results and improve known boundaries between the decidable and the undecidable cases of secondorder unification with various restrictions on secondorder variables. As a key tool we prove an undecidability result that provides a partial solution to an open problem about simultaneous rigid Eunification.